On geometric graph Ramsey numbers (Q1043821)

For graphs \(G_1, G_2, \dots, G_r\) the Ramsey number \(R(G_1, G_2, \dots, G_r)\) is the smallest positive integer \(n\) such that if the edges of a complete graph \(K_n\) are partitioned into \(r\) disjoint colour classes giving \(r\) graphs \(H_1, H_2, \dots, H_r\), then at least one \(H_i\), \(1\leq i\leq r\), contains a subgraph isomorphic to \(G_i\). A geometric graph is a graph drawn in the plane so that every vertex corresponds to a point, and every edge is a closed straight line segment connecting two vertices but not passing trough a third. A geometric graph is convex if its vertices correspond to those of a convex polygon. The geometric Ramsey number \(R_g(G_1, G_2, \dots, G_r)\) and the convex geometric Ramsey number \(R_c(G_1, G_2, \dots, G_r)\) are defined analogously to the Ramsey number. It is proved that \(R_c(C_3,C_l)=R_g(C_3,C_l)=3l-3\) for all \(l\geq 3\). A series of more general estimations on off-diagonal geometric graph Ramsey numbers are presented. The existence of large noncrossing monochromatic matchings in multicoloured geometric graphs is investigated as well.